Abstract. The determinant lower bound of Lovász, Spencer, and Vesztergombi [ European J. Combin., 7 (1986), pp. 151–160] is a general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovász, Spencer, and Vesztergombi asked if hereditary discrepancy can also be bounded from above by a function of the determinant lower bound. This was answered in the negative by Hoffman, and the largest known multiplicative gap between the two quantities for a set system of [Formula: see text] subsets of a universe of size [Formula: see text] is on the order of [Formula: see text]. On the other hand, building upon work of Matoušek [ Proc. Amer. Math. Soc., 141 (2013), pp. 451–460], Jiang and Reis [in Proceedings of the Symposium on Simplicity in Algorithms (SOSA), SIAM, Philadelphia, 2022, pp. 308–313] showed that this gap is always bounded up to constants by [Formula: see text]. This is tight when [Formula: see text] is polynomial in [Formula: see text] but leaves open the case of large [Formula: see text]. We show that the bound of Jiang and Reis is tight for nearly the entire range of [Formula: see text]. Our proof amplifies the discrepancy lower bounds of a set system derived from the discrete Haar basis via Kronecker products.

We consider the problem of partitioning n agents in an undirected social network into k almost equal in size (differing by at most one) groups, where the utility of an agent for a group is the number of her neighbors in the group. The core and envy-freeness are two compelling axiomatic fairness guarantees in such settings. The former demands that there be no coalition of agents such that each agent in the coalition has more utility for that coalition than for her own group, while the latter demands that no agent envy another agent for the group they are in. We provide (often tight) approximations to both fairness guarantees, and many of our positive results are obtained via efficient algorithms.

Many problems in computer science and applied mathematics require rounding a vector 𝐰 of fractional values lying in the interval [0,1] to a binary vector 𝐱 so that, for a given matrix 𝐀, 𝐀𝐱 is as close to 𝐀𝐰 as possible. For example, this problem arises in LP rounding algorithms used to approximate NP-hard optimization problems and in the design of uniformly distributed point sets for numerical integration. For a given matrix 𝐀, the worst-case error over all choices of 𝐰 incurred by the best possible rounding is measured by the linear discrepancy of 𝐀, a quantity studied in discrepancy theory, and introduced by Lovasz, Spencer, and Vesztergombi (EJC, 1986). We initiate the study of the computational complexity of linear discrepancy. Our investigation proceeds in two directions: (1) proving hardness results and (2) finding both exact and approximate algorithms to evaluate the linear discrepancy of certain matrices. For (1), we show that linear discrepancy is NP-hard. Thus we do not expect to find an efficient exact algorithm for the general case. Restricting our attention to matrices with a constant number of rows, we present a poly-time exact algorithm for matrices consisting of a single row and matrices with a constant number of rows and entries of bounded magnitude. We also present an exponential-time approximation algorithm for general matrices, and an algorithm that approximates linear discrepancy to within an exponential factor.